Numerical algebraic geometry and semidefinite programming

Numerical Algebraic Geometry and Algebraic Kinematics

In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometr...

Numerical Real Algebraic Geometry

Sottile’s lectures from the Oberwolfach Seminar “New trends in algorithms for real algebraic geometry”, November 23–28, 2009.

Numerical Algebraic Geometry

Ela Semidefinite Geometry of the Numerical Range

Abstract. The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular, it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI), an affine section of the semidefinite cone, is always dual to the numerical range of a matrix, which is therefore an affine projection of the semid...

The algebraic degree of semidefinite programming

Given a semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using met...